Xenharmonic Wiki

84edo: Difference between revisions

← Older edit
VisualWikitext
→Theory: expand
 
(36 intermediate revisions by 11 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|84}}
{{ED intro}}


== Theory ==
== Theory ==
84 = 7 × 12, and 84edo shares the [[3/2|perfect fifth]] with [[12edo]], [[tempering out]] the [[Pythagorean comma]] in its [[patent val]]. In the [[5-limit]] it tempers out the [[sensipent comma]]; in the [[7-limit]] [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[support]]ing [[orwell]], [[compton]], and [[sensei]].  
84edo shares the [[3/2|perfect fifth]] with [[12edo]], [[tempering out]] the [[Pythagorean comma]] in its [[patent val]]. In the [[5-limit]] it tempers out the [[sensipent comma]]; in the [[7-limit]] [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[support]]ing [[orwell]], [[compton]], and [[sensei]]. In the [[13-limit]] it is the [[optimal patent val]] for the rank-5 temperament tempering out [[144/143]].  


84edo is where the orwell temperament takes its name from, since the generator of [[7/6]] is equal to 19 steps of the edo, referencing the [[Wikipedia: Nineteen Eighty-Four|book 1984]]. Orwell in 84edo comes in two varieties – the 84e val {{val| 84 133 195 236 '''290''' }}, supporting the original orwell, and its [[patent val]] {{val| 84 133 195 236 '''291''' }} supporting [[newspeak]]. 84edo orwell offers [[mos scale]]s of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale.
84edo is where the orwell temperament takes its name from, since the generator of [[7/6]] is equal to 19 steps of the edo, referencing the [[Wikipedia: Nineteen Eighty-Four|book 1984]]. Orwell in 84edo comes in two varieties—the 84e val {{val| 84 133 195 236 '''290''' }}, supporting the original orwell, and its [[patent val]] {{val| 84 133 195 236 '''291''' }} supporting [[newspeak]]. 84edo orwell offers [[mos scale]]s of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale.
 
It has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], and [[31/1|31]]. In fact, it is [[consistent]] to the no-11 no-17 [[25-odd-limit]]. In the [[13-limit]] it is the [[optimal patent val]] for the rank-5 temperament tempering out [[144/143]].  


=== Prime harmonics ===
=== Prime harmonics ===
Line 17: Line 15:


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
! #
! #
! Cents
! Cents
! Approximate Ratios*
! Approximate ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| [[1/1]]
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 31: Line 30:
|-
|-
| 1
| 1
| 14.286
| 14.3
| ''81/80'', 105/104, 126/125, 169/168, 196/195
| ''[[81/80]]'', [[105/104]], [[126/125]], [[169/168]], [[196/195]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 38: Line 37:
|-
|-
| 2
| 2
| 28.571
| 28.6
| 50/49, 64/63, 65/64, ''91/90''
| [[50/49]], [[64/63]], [[65/64]], ''[[91/90]]''
| Dup 1sn
| Dup 1sn
| ^^1
| ^^1
Line 45: Line 44:
|-
|-
| 3
| 3
| 42.857
| 42.9
| 36/35, 40/39, 46/45, 49/48
| [[36/35]], [[40/39]], [[46/45]], [[49/48]]
| Trup 1sn
| Trup 1sn
| ^^^1
| ^^^1
Line 52: Line 51:
|-
|-
| 4
| 4
| 57.143
| 57.1
| ''27/26''
| ''[[27/26]]''
| Trudminor 2nd
| Trudminor 2nd
| vvvm2
| vvvm2
Line 59: Line 58:
|-
|-
| 5
| 5
| 71.429
| 71.4
| 24/23, 25/24, 26/25, ''28/27''
| [[24/23]], [[25/24]], [[26/25]], ''[[28/27]]''
| Dudminor 2nd
| Dudminor 2nd
| vvm2
| vvm2
Line 66: Line 65:
|-
|-
| 6
| 6
| 85.714
| 85.7
| 20/19, 21/20
| [[20/19]], [[21/20]]
| Downminor 2nd
| Downminor 2nd
| vm2
| vm2
Line 73: Line 72:
|-
|-
| 7
| 7
| 100.000
| 100.0
| 19/18
| [[19/18]]
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 80: Line 79:
|-
|-
| 8
| 8
| 114.286
| 114.3
| 15/14, 16/15
| [[15/14]], [[16/15]]
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
Line 87: Line 86:
|-
|-
| 9
| 9
| 128.571
| 128.6
| 14/13
| [[14/13]]
| Dupminor 2nd
| Dupminor 2nd
| ^^m2
| ^^m2
Line 94: Line 93:
|-
|-
| 10
| 10
| 142.857
| 142.9
| 13/12
| [[13/12]]
| Trupminor 2nd
| Trupminor 2nd
| ^^^m2
| ^^^m2
Line 101: Line 100:
|-
|-
| 11
| 11
| 157.143
| 157.1
| 23/21
| [[23/21]]
| Trudmajor 2nd
| Trudmajor 2nd
| vvvM2
| vvvM2
Line 108: Line 107:
|-
|-
| 12
| 12
| 171.429
| 171.4
| 21/19
| [[21/19]]
| Dudmajor 2nd
| Dudmajor 2nd
| vvM2
| vvM2
Line 115: Line 114:
|-
|-
| 13
| 13
| 185.714
| 185.7
| 10/9
| [[10/9]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
Line 122: Line 121:
|-
|-
| 14
| 14
| 200.000
| 200.0
| 9/8
| [[9/8]]
| Major 2nd
| Major 2nd
| M2
| M2
Line 129: Line 128:
|-
|-
| 15
| 15
| 214.286
| 214.3
| 26/23
| [[26/23]]
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 136: Line 135:
|-
|-
| 16
| 16
| 228.571
| 228.6
| 8/7
| [[8/7]]
| Dupmajor 2nd
| Dupmajor 2nd
| ^^M2
| ^^M2
Line 143: Line 142:
|-
|-
| 17
| 17
| 242.857
| 242.9
| 15/13, 23/20
| [[15/13]], [[23/20]]
| Trupmajor 2nd
| Trupmajor 2nd
| ^^^M2
| ^^^M2
Line 150: Line 149:
|-
|-
| 18
| 18
| 257.143
| 257.1
| 52/45
| [[52/45]]
| Trudminor 3rd
| Trudminor 3rd
| vvvm3
| vvvm3
Line 157: Line 156:
|-
|-
| 19
| 19
| 271.429
| 271.4
| 7/6
| [[7/6]]
| Dudminor 3rd
| Dudminor 3rd
| vvm2
| vvm2
Line 164: Line 163:
|-
|-
| 20
| 20
| 285.714
| 285.7
| 45/38, 46/39
| [[45/38]], [[46/39]]
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 171: Line 170:
|-
|-
| 21
| 21
| 300.000
| 300.0
| 19/16, 25/21, 32/27
| [[19/16]], [[25/21]], [[32/27]]
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 178: Line 177:
|-
|-
| 22
| 22
| 314.286
| 314.3
| 6/5
| [[6/5]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
Line 185: Line 184:
|-
|-
| 23
| 23
| 328.571
| 328.6
| 23/19
| [[23/19]]
| Dupminor 3rd
| Dupminor 3rd
| ^^m3
| ^^m3
Line 192: Line 191:
|-
|-
| 24
| 24
| 342.857
| 342.9
| 28/23, 39/32
| [[28/23]], [[39/32]]
| Trupminor 3rd
| Trupminor 3rd
| ^^^m3
| ^^^m3
Line 199: Line 198:
|-
|-
| 25
| 25
| 357.143
| 357.1
| 16/13
| [[16/13]]
| Trudmajor 3rd
| Trudmajor 3rd
| vvvM3
| vvvM3
Line 206: Line 205:
|-
|-
| 26
| 26
| 371.429
| 371.4
| 26/21
| [[26/21]]
| Dudmajor 3rd
| Dudmajor 3rd
| vvM3
| vvM3
Line 213: Line 212:
|-
|-
| 27
| 27
| 385.714
| 385.7
| 5/4
| [[5/4]]
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
Line 220: Line 219:
|-
|-
| 28
| 28
| 400.000
| 400.0
| 24/19
| [[24/19]]
| Major 3rd
| Major 3rd
| M3
| M3
Line 227: Line 226:
|-
|-
| 29
| 29
| 414.286
| 414.3
| 19/15
| [[19/15]]
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
Line 234: Line 233:
|-
|-
| 30
| 30
| 428.571
| 428.6
| 9/7, 23/18, 32/25
| [[9/7]], [[23/18]], [[32/25]]
| Dupmajor 3rd
| Dupmajor 3rd
| ^^M3
| ^^M3
Line 241: Line 240:
|-
|-
| 31
| 31
| 442.857
| 442.9
| 84/65
| [[84/65]]
| Trupmajor 3rd
| Trupmajor 3rd
| ^^^M3
| ^^^M3
Line 248: Line 247:
|-
|-
| 32
| 32
| 457.143
| 457.1
| 13/10, 30/23
| [[13/10]], [[30/23]]
| Trud 4th
| Trud 4th
| vvv4
| vvv4
Line 255: Line 254:
|-
|-
| 33
| 33
| 471.429
| 471.4
| 21/16
| [[21/16]]
| Dud 4th
| Dud 4th
| vv4
| vv4
Line 262: Line 261:
|-
|-
| 34
| 34
| 485.714
| 485.7
| 65/49
| [[65/49]]
| Down 4th
| Down 4th
| v4
| v4
Line 269: Line 268:
|-
|-
| 35
| 35
| 500.000
| 500.0
| 4/3
| [[4/3]]
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 276: Line 275:
|-
|-
| 36
| 36
| 514.286
| 514.3
| 27/20
| [[27/20]]
| Up 4th
| Up 4th
| ^4
| ^4
Line 283: Line 282:
|-
|-
| 37
| 37
| 528.571
| 528.6
| 19/14
| [[19/14]]
| Dup 4th
| Dup 4th
| ^^4
| ^^4
Line 290: Line 289:
|-
|-
| 38
| 38
| 542.857
| 542.9
| 26/19
| [[26/19]]
| Trup 4th
| Trup 4th
| ^^^4
| ^^^4
Line 297: Line 296:
|-
|-
| 39
| 39
| 557.143
| 557.1
| 18/13
| [[18/13]]
| Trudaug 4th
| Trudaug 4th
| vvvA4
| vvvA4
Line 304: Line 303:
|-
|-
| 40
| 40
| 571.429
| 571.4
| 25/18, 32/23
| [[25/18]], [[32/23]]
| Dudaug 4th
| Dudaug 4th
| vvA4
| vvA4
Line 311: Line 310:
|-
|-
| 41
| 41
| 585.714
| 585.7
| 7/5
| [[7/5]]
| Downaug 4th
| Downaug 4th
| vA4
| vA4
Line 318: Line 317:
|-
|-
| 42
| 42
| 600.000
| 600.0
| 27/19, 38/27
| [[27/19]], [[38/27]]
| Aug 4th, Dim 5th
| Aug 4th, Dim 5th
| A4, d5
| A4, d5
Line 325: Line 324:
|-
|-
| 43
| 43
| 614.286
| 614.3
| 10/7
| [[10/7]]
| Updim 5th
| Updim 5th
| ^d5
| ^d5
Line 332: Line 331:
|-
|-
| 44
| 44
| 628.571
| 628.6
| 23/16, 36/25
| [[23/16]], [[36/25]]
| Dupdim 5th
| Dupdim 5th
| ^^d5
| ^^d5
Line 339: Line 338:
|-
|-
| 45
| 45
| 642.857
| 642.9
| 13/9
| [[13/9]]
| Trupdim 5th
| Trupdim 5th
| ^^^d5
| ^^^d5
Line 346: Line 345:
|-
|-
| 46
| 46
| 657.143
| 657.1
| 19/13
| [[19/13]]
| Trud 5th
| Trud 5th
| vvv5
| vvv5
Line 353: Line 352:
|-
|-
| 47
| 47
| 671.429
| 671.4
| 28/19
| [[28/19]]
| Dud 5th
| Dud 5th
| vv5
| vv5
Line 360: Line 359:
|-
|-
| 48
| 48
| 685.714
| 685.7
| 40/27
| [[40/27]]
| Down 5th
| Down 5th
| v5
| v5
Line 367: Line 366:
|-
|-
| 49
| 49
| 700.000
| 700.0
| 3/2
| [[3/2]]
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 374: Line 373:
|-
|-
| 50
| 50
| 714.286
| 714.3
| 98/65
| [[98/65]]
| Up 5th
| Up 5th
| ^5
| ^5
Line 381: Line 380:
|-
|-
| 51
| 51
| 728.571
| 728.6
| 32/21
| [[32/21]]
| Dup 5th
| Dup 5th
| ^^5
| ^^5
Line 388: Line 387:
|-
|-
| 52
| 52
| 742.857
| 742.9
| 20/13, 23/15
| [[20/13]], [[23/15]]
| Trup 5th
| Trup 5th
| ^^^5
| ^^^5
Line 395: Line 394:
|-
|-
| 53
| 53
| 757.143
| 757.1
| 65/42
| [[65/42]]
| Trudminor 6th
| Trudminor 6th
| vvvm6
| vvvm6
Line 402: Line 401:
|-
|-
| 54
| 54
| 771.429
| 771.4
| 14/9, 25/16, 36/23
| [[14/9]], [[25/16]], [[36/23]]
| Dudminor 6th
| Dudminor 6th
| vvm6
| vvm6
Line 409: Line 408:
|-
|-
| 55
| 55
| 785.714
| 785.7
| 30/19
| [[30/19]]
| Downminor 6th
| Downminor 6th
| vm6
| vm6
Line 416: Line 415:
|-
|-
| 56
| 56
| 800.000
| 800.0
| 19/12
| [[19/12]]
| Minor 6th
| Minor 6th
| m6
| m6
Line 423: Line 422:
|-
|-
| 57
| 57
| 814.286
| 814.3
| 8/5
| [[8/5]]
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
Line 430: Line 429:
|-
|-
| 58
| 58
| 828.571
| 828.6
| 21/13
| [[21/13]]
| Dupminor 6th
| Dupminor 6th
| ^^m6
| ^^m6
Line 437: Line 436:
|-
|-
| 59
| 59
| 842.857
| 842.9
| 13/8
| [[13/8]]
| Trupminor 6th
| Trupminor 6th
| ^^^m6
| ^^^m6
Line 444: Line 443:
|-
|-
| 60
| 60
| 857.143
| 857.1
| 23/14, 64/39
| [[23/14]], [[64/39]]
| Trudmajor 6th
| Trudmajor 6th
| vvvM6
| vvvM6
Line 451: Line 450:
|-
|-
| 61
| 61
| 871.429
| 871.4
| 38/23
| [[38/23]]
| Dudmajor 6th
| Dudmajor 6th
| vvM6
| vvM6
Line 458: Line 457:
|-
|-
| 62
| 62
| 885.714
| 885.7
| 5/3
| [[5/3]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
Line 465: Line 464:
|-
|-
| 63
| 63
| 900.000
| 900.0
| 32/19, 27/16, 42/25
| [[32/19]], [[27/16]], [[42/25]]
| Major 6th
| Major 6th
| M6
| M6
Line 472: Line 471:
|-
|-
| 64
| 64
| 914.286
| 914.3
| 39/23, 76/45
| [[39/23]], [[76/45]]
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 479: Line 478:
|-
|-
| 65
| 65
| 928.571
| 928.6
| 12/7
| [[12/7]]
| Dupmajor 6th
| Dupmajor 6th
| ^^M6
| ^^M6
Line 486: Line 485:
|-
|-
| 66
| 66
| 942.857
| 942.9
| 45/26
| [[45/26]]
| Trupmajor 6th
| Trupmajor 6th
| ^^^M6
| ^^^M6
Line 493: Line 492:
|-
|-
| 67
| 67
| 957.143
| 957.1
| 26/15, 40/23
| [[26/15]], [[40/23]]
| Trudminor 7th
| Trudminor 7th
| vvvm7
| vvvm7
Line 500: Line 499:
|-
|-
| 68
| 68
| 971.429
| 971.4
| 7/4
| [[7/4]]
| Dudminor 7th
| Dudminor 7th
| vvm7
| vvm7
Line 507: Line 506:
|-
|-
| 69
| 69
| 985.714
| 985.7
| 23/13
| [[23/13]]
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 514: Line 513:
|-
|-
| 70
| 70
| 1000.000
| 1000.0
| 16/9
| [[16/9]]
| Minor 7th
| Minor 7th
| m7
| m7
Line 521: Line 520:
|-
|-
| 71
| 71
| 1014.286
| 1014.3
| 9/5
| [[9/5]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
Line 528: Line 527:
|-
|-
| 72
| 72
| 1028.571
| 1028.6
| 38/21
| [[38/21]]
| Dupminor 7th
| Dupminor 7th
| ^^m7
| ^^m7
Line 535: Line 534:
|-
|-
| 73
| 73
| 1042.857
| 1042.9
| 42/23
| [[42/23]]
| Trupminor 7th
| Trupminor 7th
| ^^^m7
| ^^^m7
Line 542: Line 541:
|-
|-
| 74
| 74
| 1057.143
| 1057.1
| 24/13
| [[24/13]]
| Trudmajor 7th
| Trudmajor 7th
| vvvM7
| vvvM7
Line 549: Line 548:
|-
|-
| 75
| 75
| 1071.429
| 1071.4
| 13/7
| [[13/7]]
| Dudmajor 7th
| Dudmajor 7th
| vvM7
| vvM7
Line 556: Line 555:
|-
|-
| 76
| 76
| 1085.714
| 1085.7
| 15/8, 28/15
| [[15/8]], [[28/15]]
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
Line 563: Line 562:
|-
|-
| 77
| 77
| 1100.000
| 1100.0
| 36/19
| [[36/19]]
| Major 7th
| Major 7th
| M7
| M7
Line 570: Line 569:
|-
|-
| 78
| 78
| 1114.286
| 1114.3
| 19/10, 40/21
| [[19/10]], [[40/21]]
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
Line 577: Line 576:
|-
|-
| 79
| 79
| 1128.571
| 1128.6
| 23/12, 25/13, ''27/14'', 48/25
| [[23/12]], [[25/13]], ''[[27/14]]'', [[48/25]]
| Dupmajor 7th
| Dupmajor 7th
| ^^M7
| ^^M7
Line 584: Line 583:
|-
|-
| 80
| 80
| 1142.857
| 1142.9
| ''52/27''
| ''[[52/27]]''
| Trupmajor 7th
| Trupmajor 7th
| ^^^M7
| ^^^M7
Line 591: Line 590:
|-
|-
| 81
| 81
| 1157.143
| 1157.1
| 35/18, 39/20, 96/49
| [[35/18]], [[39/20]], [[96/49]]
| Trud 8ve
| Trud 8ve
| vvv8
| vvv8
Line 598: Line 597:
|-
|-
| 82
| 82
| 1171.429
| 1171.4
| 45/23, 49/25, 63/32, 128/65, ''180/91''
| [[45/23]], [[49/25]], [[63/32]], [[128/65]], ''[[180/91]]''
| Dud 8ve
| Dud 8ve
| vv8
| vv8
Line 605: Line 604:
|-
|-
| 83
| 83
| 1185.714
| 1185.7
| 125/63, ''160/81'', 195/98, 336/169
| [[125/63]], ''[[160/81]]'', [[195/98]], [[336/169]]
| Down 8ve
| Down 8ve
| v8
| v8
Line 612: Line 611:
|-
|-
| 84
| 84
| 1200.000
| 1200.0
| 2/1
| [[2/1]]
| Perfect 8ve
| Perfect 8ve
| P8
| P8
| D
| D
|}
|}
<nowiki>*</nowiki> as a 2.3.5.7.13.19.23-subgroup temperament
<nowiki/>* As a 2.3.5.7.13.19.23-subgroup temperament


== Notation ==
== Notation ==
=== Ups and downs notation ===
84edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Ups and downs sharpness|84}}
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7|84}}
=== 4L 5s (gramitonic) notation ===
=== 4L 5s (gramitonic) notation ===
The notation of Orwell[9]. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & "amp" and @ "at".  
This notation is based on Orwell[9]. Notes are denoted as {{nowrap|LsLsLsLss {{=}} JKLMNOPQRJ}}, and raising and lowering by a chroma ({{nowrap|L − s}}), 3 steps in this instance, is denoted by &amp;&nbsp;("amp") and @&nbsp;("at").  


{| class="wikitable center-1 right-2 center-3"
{| class="wikitable center-1 right-2 center-3"
Line 740: Line 747:
| 2/1
| 2/1
|}
|}
== Approximation to JI ==
=== 15-odd-limit intervals ===
{{Q-odd-limit intervals|84}}
=== Higher-limit JI ===
84edo has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], [[31/1|31]], 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit, with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics. In fact, it is [[consistent]] in the no-11 no-17 no-27 no-37 no-47 no-49 no-51 no-55 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~0.13{{cent}} (off by ~7.3{{cent}}), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too when supported by context. The only missing primes are thus 17, 37, 47, 67, 71, 79 and 83, which except for 17 are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result).


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 754: Line 769:
| 2.3.5
| 2.3.5
| 78732/78125, 531441/524288
| 78732/78125, 531441/524288
| {{mapping| 84 133 195 }}
| {{Mapping| 84 133 195 }}
| +0.498
| +0.498
| 0.531
| 0.531
Line 761: Line 776:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1728/1715, 78732/78125
| 225/224, 1728/1715, 78732/78125
| {{mapping| 84 133 195 236 }}
| {{Mapping| 84 133 195 236 }}
| +0.141
| +0.141
| 0.769
| 0.769
| 5.39
| 5.39
|-
|-
| 2.3.5.7.13
| 225/224, 351/350, 640/637, 1701/1690
| {{Mapping| 84 133 195 236 311 }}
| −0.013
| 0.754
| 5.28
|- style="border-top: double;"
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 441/440, 1344/1331, 1728/1715
| 225/224, 441/440, 1344/1331, 1728/1715
| {{mapping| 84 133 195 236 291 }} (84)
| {{Mapping| 84 133 195 236 291 }} (84)
| -0.225
| −0.225
| 1.003
| 1.003
| 7.02
| 7.02
|-
|-
| 2.3.5.7.11.13
| 144/143, 225/224, 351/350, 441/440, 975/968
| {{Mapping| 84 133 195 236 291 311 }} (84)
| −0.292
| 0.928
| 6.50
|- style="border-top: double;"
| 2.3.5.7.11
| 2.3.5.7.11
| 99/98, 121/120, 176/175, 78732/78125
| 99/98, 121/120, 176/175, 78732/78125
| {{mapping| 84 133 195 236 290 }} (84e)
| {{Mapping| 84 133 195 236 290 }} (84e)
| +0.601
| +0.601
| 1.151
| 1.151
| 8.05
| 8.05
|-
| 2.3.5.7.11.13
| 99/98, 121/120, 176/175, 275/273, 1701/1690
| {{Mapping| 84 133 195 236 290 311 }} (84e)
| +0.396
| 1.146
| 8.02
|}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperament
|-
|-
| rowspan="2" | 1
| 1
| rowspan="2" | 19\84
| 19\84
| rowspan="2" | 271.43
| 271.4
| rowspan="2" | 7/6
| 7/6
| [[Orwell]] (84e)
| [[Orwell]] (84e) / [[newspeak]] (84)
|-
| [[Newspeak]] (84)
|-
|-
| 1
| 1
| 25\84
| 25\84
| 357.14
| 357.1
| 768/625
| 768/625
| [[Dodifo]]
| [[Dodifo]]
Line 805: Line 841:
| 1
| 1
| 27\84
| 27\84
| 385.71
| 385.7
| 5/4
| 5/4
| [[Mutt]]
| [[Mutt]]
Line 811: Line 847:
| 1
| 1
| 31\84
| 31\84
| 442.86
| 442.9
| 125/81
| 162/125
| [[Sensei]]
| [[Sensei]]
|-
|-
| 1
| 1
| 41\84
| 41\84
| 585.71
| 585.7
| 7/5
| 7/5
| [[Merman]]
| [[Merman]]
Line 823: Line 859:
| 2
| 2
| 5\84
| 5\84
| 71.43
| 71.4
| 25/24
| 25/24
| [[Narayana]]
| [[Narayana]]
Line 829: Line 865:
| 2
| 2
| 11\84
| 11\84
| 157.14
| 157.1
| 35/32
| 35/32
| [[Bison]]
| [[Bison]]
Line 835: Line 871:
| 2
| 2
| 13\84
| 13\84
| 185.71
| 185.7
| 10/9
| 10/9
| [[Secant]]
| [[Secant]]
Line 841: Line 877:
| 3
| 3
| 11\84
| 11\84
| 157.14
| 157.1
| 35/32
| 35/32
| [[Nessafof]]
| [[Nessafof]]
|-
| 6
| 5\84
| 71.4
| 25/24
| [[Trivish]]
|-
|-
| 7
| 7
| 5\84
| 5\84
| 500.00<br>(14.29)
| 14.3
| 4/3<br>(81/80)
| 81/80
| [[Absurdity]]
| [[Absurdity]]
|-
|-
| 12
| 12
| 27\84<br>(1\84)
| 1\84
| 385.71<br>(14.29)
| 14.3
| 5/4<br>(126/125)
| 126/125
| [[Compton]]
| [[Compton]]
|-
| 12
| 2\84
| 28.6
| 64/63
| [[Catler]] (84c)
|-
|-
| 21
| 21
| 41\84<br>(1\84)
| 1\84)
| 585.71<br>(14.29)
| 14.3
| 91875/65536<br>(126/125)
| 126/125
| [[Akjayland]]
| [[Akjayland]]
|-
|-
| 28
| 28
| 49\84<br>(1\84)
| 1\84
| 500.00<br>(14.29)
| 14.3
| 4/3<br>(105/104)
| 105/104
| [[Oquatonic]]
| [[Oquatonic]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Scales ==
== Scales ==
=== MOS ===
=== MOS ===
Brightest mode is listed.
Brightest mode is listed.


* [[Orwell]]
* [[Orwell]]
** Orwell[9], [[4L 5s]] - 11 8 11 8 11 8 11 8 8  
** Orwell[9] ([[4L 5s]]) – 11 8 11 8 11 8 11 8 8  
** Orwell[13] - [[9L 4s]] - 8 8 8 3 8 8 3 8 8 3 8 8 3
** Orwell[13] ([[9L 4s]]) – 8 8 8 3 8 8 3 8 8 3 8 8 3
** Orwell[22] - [[13L 9s]]
** Orwell[22] ([[13L 9s]])
** Orwell[31] - [[22L 9s]]
** Orwell[31] ([[22L 9s]])


=== Other ===
=== Other ===
* [[5- to 10-tone scales in 84edo]]
* [[5- to 10-tone scales in 84edo]]
* [[Maeve Gutierrez|Gutierrez Moonglade scale]]
== Instruments ==
If you have a precise enough tuner and stable enough instruments, 84edo can be played using 7 instruments tuned a 14th of a tone apart.
You could also try the [[Lumatone mapping for 84edo]]


== Music ==
== Music ==
Line 891: Line 944:
* ''Two5'' for tenor trombone and piano (1991) [https://youtu.be/YOtQZIqfY1w Fulkerson &amp; Denyer recording (YouTube)]
* ''Two5'' for tenor trombone and piano (1991) [https://youtu.be/YOtQZIqfY1w Fulkerson &amp; Denyer recording (YouTube)]
* ''Two6'' for violin and piano (1992) [https://youtu.be/XkX37zH6AbU Haar &amp; Snijders recording (YouTube)]
* ''Two6'' for violin and piano (1992) [https://youtu.be/XkX37zH6AbU Haar &amp; Snijders recording (YouTube)]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/Sqkxrmwggr0 ''microtonal improvisation in 84edo''] (2025)
* [https://www.youtube.com/shorts/Qu6UIA2NmmQ ''84edo groove''] (2026)


; [[Eliora]]
; [[Eliora]]
Retrieved from "https://en.xen.wiki/w/84edo"